Trigonometric ratios of any angle

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Trigonometric ratios of all quadrants

Let a line OP start from OX revolve round O in the anticlockwise direction. Then it will trace positive angle XOP. If OP is in the first quadrant, then \(\angle\)XOP lies between 0° and 90°. If OP is in the second quadrant, then \(\angle\)XOP lies between 90° and 180°. Similarly, when OP is in the third or the fourth quadrant, then \(\angle\)XOP lies between 180° and 270° or 270° and 360° respectively.If OP revolves in the clockwise direction, the angle traced is negative.

Rule of signs

In the above figure, lines measured along OX' or OY' are considered positive. Lines measured along OX and OY are considered negative. If PM is perpendicular to XX', then

Both OM and MP are positive when OP is in the first quadrant.

OM is negative and MP is positive when OP is in the second quadrant.

Both OM and MP are negative when OP is in the third quadrant.

OM is positive and MP is negative when OP is in the fourth quadrant.

The line OP is always positive.

Thus the values of trigonometric ratios in different quadrants are as follows:-

1st quadrant

2nd quadrant

3rd quadrant

4th quadrant

all the ratios are (+e) positive

sin and cosec (+ve) remaining (-ve)

tan and cot (+ve) remaining (-ve)

cos and sec (+ve) remaining (-v)

Ratios of (90° - A)

Let revolving line OP start from OX and trace out an angle XOP = A. Let another line OQ (= OP) revolve from OX to OYn and return back to OQ through an angle YOQ = A.